A Lie-theoretic Construction of Cartan-Moser Chains

Let $M^3 \subset \mathbb{C}^2$ be a $\mathcal{C}^\omega$ Levi nondegenerate hypersurface. In the literature, Cartan-Moser chains are detected from rather advanced considerations: either from the construction of a Cartan connection associated with the CR equivalence problem; or from the construction of a formal or converging Poincar\'e-Moser normal form. This note provides an alternative direct elementary construction, based on the inspection of the Lie prolongations of $5$ infinitesimal holomorphic automorphisms to the space of second order jets of CR-transversal curves. Within the $4$-dimensional jet fiber, the orbits of these $5$ prolonged fields happen to have a simple cubic $2$-dimensional degenerate exceptional orbit, the chain locus: $$\Sigma_0 \,:=\, \big\{ (x_1,y_1,x_2,y_2) \in \mathbb{R}^4 \colon\,\, x_2 = -2x_1^2y_1-2y_1^3,\,\,\, y_2 = 2x_1y_1^2 + 2x_1^3 \big\}.$$ By plain translations, we may capture all points by working only at one point, the origin, and computations, although conceptually enlightening, become disappointingly simple.